According to Boris Golubov, BV functions of a single variable were first introduced by Camille Jordan, in the paper (Jordan 1881) dealing with the convergence of Fourier series. The first successful step in the generalization of this concept to functions of several variables was due to Leonida Tonelli, who introduced a class of continuous BV functions in 1926 (Cesari 1986, pp. 47–48), to extend his direct method for finding solutions to problems in the calculus of variations in more than one variable. Ten years after, in (Cesari 1936), Lamberto Cesari changed the continuity requirement in Tonelli's definition to a less restrictive integrability requirement, obtaining for the first time the class of functions of bounded variation of several variables in its full generality: as Jordan did before him, he applied the concept to resolve of a problem concerning the convergence of Fourier series, but for functions of two variables. After him, several authors applied BV functions to study Fourier series in several variables, geometric measure theory, calculus of variations, and mathematical physics. Renato Caccioppoli and Ennio de Giorgi used them to define measure of nonsmooth boundaries of sets (see the entry "Caccioppoli set" for further information). Olga Arsenievna Oleinik introduced her view of generalized solutions for nonlinear partial differential equations as functions from the space BV in the paper (Oleinik 1957), and was able to construct a generalized solution of bounded variation of a first order partial differential equation in the paper (Oleinik 1959): few years later, Edward D. Conway and Joel A. Smoller applied BVfunctions to the study of a single nonlinear hyperbolic partial differential equation of first order in the paper (Conway & Smoller 1966), proving that the solution of the Cauchy problem for such equations is a function of bounded variation, provided the initial value belongs to the same class. Aizik Isaakovich Vol'pert developed extensively a calculus for BV functions: in the paper (Vol'pert 1967) he proved the chain rule for BV functions and in the book (Hudjaev & Vol'pert 1985) he, jointly with his pupil Sergei Ivanovich Hudjaev, explored extensively the properties of BV functions and their application. His chain rule formula was later extended by Luigi Ambrosio and Gianni Dal Maso in the paper (Ambrosio & Dal Maso 1990).
Definition 1.1. The total variation of a realvalued (or more generally complexvalued) function f, defined on an interval [a, b]⊂ℝ is the quantity
V
a
b
(
f
)
=
sup
P
∈
P
∑
i
=
0
n
P
−
1

f
(
x
i
+
1
)
−
f
(
x
i
)

.
where the supremum is taken over the set
P
=
{
P
=
{
x
0
,
…
,
x
n
P
}

P
is a partition of
[
a
,
b
]
}
of all partitions of the interval considered.
If f is differentiable and its derivative is Riemannintegrable, its total variation is the vertical component of the arclength of its graph, that is to say,
V
a
b
(
f
)
=
∫
a
b

f
′
(
x
)

d
x
.
Definition 1.2. A realvalued function
f
on the real line is said to be of bounded variation (BV function) on a chosen interval [a, b]⊂ℝ if its total variation is finite, i.e.
f
∈
B
V
(
[
a
,
b
]
)
⟺
V
a
b
(
f
)
<
+
∞
It can be proved that a real function ƒ is of bounded variation in
[
a
,
b
]
if and only if it can be written as the difference ƒ = ƒ_{1} − ƒ_{2} of two nondecreasing functions on
[
a
,
b
]
: this result is known as the Jordan decomposition of a function and it is related to the Jordan decomposition of a measure.
Through the Stieltjes integral, any function of bounded variation on a closed interval [a, b] defines a bounded linear functional on C([a, b]). In this special case, the Riesz representation theorem states that every bounded linear functional arises uniquely in this way. The normalised positive functionals or probability measures correspond to positive nondecreasing lower semicontinuous functions. This point of view has been important in spectral theory, in particular in its application to ordinary differential equations.
Functions of bounded variation, BV functions, are functions whose distributional derivative is a finite Radon measure. More precisely:
Definition 2.1. Let
Ω
be an open subset of ℝ^{n}. A function
u
belonging to
L
1
(
Ω
)
is said of bounded variation (BV function), and written
u
∈
B
V
(
Ω
)
if there exists a finite vector Radon measure
D
u
∈
M
(
Ω
,
R
n
)
such that the following equality holds
∫
Ω
u
(
x
)
d
i
v
ϕ
(
x
)
d
x
=
−
∫
Ω
⟨
ϕ
,
D
u
(
x
)
⟩
∀
ϕ
∈
C
c
1
(
Ω
,
R
n
)
that is,
u
defines a linear functional on the space
C
c
1
(
Ω
,
R
n
)
of continuously differentiable vector functions
ϕ
of compact support contained in
Ω
: the vector measure
D
u
represents therefore the distributional or weak gradient of
u
.
An equivalent definition is the following.
Definition 2.2. Given a function
u
belonging to
L
1
(
Ω
)
, the total variation of
u
in
Ω
is defined as
V
(
u
,
Ω
)
:=
sup
{
∫
Ω
u
(
x
)
d
i
v
ϕ
(
x
)
d
x
:
ϕ
∈
C
c
1
(
Ω
,
R
n
)
,
∥
ϕ
∥
L
∞
(
Ω
)
≤
1
}
where
∥
∥
L
∞
(
Ω
)
is the essential supremum norm. Sometimes, especially in the theory of Caccioppoli sets, the following notation is used
∫
Ω

D
u

=
V
(
u
,
Ω
)
in order to emphasize that
V
(
u
,
Ω
)
is the total variation of the distributional / weak gradient of
u
. This notation reminds also that if
u
is of class
C
1
(i.e. a continuous and differentiable function having continuous derivatives) then its variation is exactly the integral of the absolute value of its gradient.
The space of functions of bounded variation (BV functions) can then be defined as
B
V
(
Ω
)
=
{
u
∈
L
1
(
Ω
)
:
V
(
u
,
Ω
)
<
+
∞
}
The two definitions are equivalent since if
V
(
u
,
Ω
)
<
+
∞
then

∫
Ω
u
(
x
)
d
i
v
ϕ
(
x
)
d
x

≤
V
(
u
,
Ω
)
∥
ϕ
∥
L
∞
(
Ω
)
∀
ϕ
∈
C
c
1
(
Ω
,
R
n
)
therefore
ϕ
↦
∫
Ω
u
(
x
)
d
i
v
ϕ
(
x
)
d
x
defines a continuous linear functional on the space
C
c
1
(
Ω
,
R
n
)
. Since
C
c
1
(
Ω
,
R
n
)
⊂
C
0
(
Ω
,
R
n
)
as a linear subspace, this continuous linear functional can be extended continuously and linearily to the whole
C
0
(
Ω
,
R
n
)
by the Hahn–Banach theorem. Hence the continuous linear functional defines a Radon measure by the RieszMarkov Theorem.
If the function space of locally integrable functions, i.e. functions belonging to
L
l
o
c
1
(
Ω
)
, is considered in the preceding definitions 1.2, 2.1 and 2.2 instead of the one of globally integrable functions, then the function space defined is that of functions of locally bounded variation. Precisely, developing this idea for definition 2.2, a local variation is defined as follows,
V
(
u
,
U
)
:=
sup
{
∫
Ω
u
(
x
)
d
i
v
ϕ
(
x
)
d
x
:
ϕ
∈
C
c
1
(
U
,
R
n
)
,
∥
ϕ
∥
L
∞
(
Ω
)
≤
1
}
for every set
U
∈
O
c
(
Ω
)
, having defined
O
c
(
Ω
)
as the set of all precompact open subsets of
Ω
with respect to the standard topology of finitedimensional vector spaces, and correspondingly the class of functions of locally bounded variation is defined as
B
V
l
o
c
(
Ω
)
=
{
u
∈
L
l
o
c
1
(
Ω
)
:
V
(
u
,
U
)
<
+
∞
∀
U
∈
O
c
(
Ω
)
}
There are basically two distinct conventions for the notation of spaces of functions of locally or globally bounded variation, and unfortunately they are quite similar: the first one, which is the one adopted in this entry, is used for example in references Giusti (1984) (partially), Hudjaev & Vol'pert (1985) (partially), Giaquinta, Modica & Souček (1998) and is the following one
B
V
(
Ω
)
identifies the space of functions of globally bounded variation
B
V
l
o
c
(
Ω
)
identifies the space of functions of locally bounded variation
The second one, which is adopted in references Vol'pert (1967) and Maz'ya (1985) (partially), is the following:
B
V
¯
(
Ω
)
identifies the space of functions of globally bounded variation
B
V
(
Ω
)
identifies the space of functions of locally bounded variation
Only the properties common to functions of one variable and to functions of several variables will be considered in the following, and proofs will be carried on only for functions of several variables since the proof for the case of one variable is a straightforward adaptation of the several variables case: also, in each section it will be stated if the property is shared also by functions of locally bounded variation or not. References (Giusti 1984, pp. 7–9), (Hudjaev & Vol'pert 1985) and (Màlek et al. 1996) are extensively used.
In the case of one variable, the assertion is clear: for each point
x
0
in the interval
[
a
,
b
]
⊂ℝ of definition of the function
u
, either one of the following two assertions is true
lim
x
→
x
0
−
u
(
x
)
=
lim
x
→
x
0
+
u
(
x
)
lim
x
→
x
0
−
u
(
x
)
≠
lim
x
→
x
0
+
u
(
x
)
while both limits exist and are finite. In the case of functions of several variables, there are some premises to understand: first of all, there is a continuum of directions along which it is possible to approach a given point
x
0
belonging to the domain
Ω
⊂ℝ^{n}. It is necessary to make precise a suitable concept of limit: choosing a unit vector
a
^
∈
R
n
it is possible to divide
Ω
in two sets
Ω
(
a
^
,
x
0
)
=
Ω
∩
{
x
∈
R
n

⟨
x
−
x
0
,
a
^
⟩
>
0
}
Ω
(
−
a
^
,
x
0
)
=
Ω
∩
{
x
∈
R
n

⟨
x
−
x
0
,
−
a
^
⟩
>
0
}
Then for each point
x
0
belonging to the domain
Ω
∈
R
n
of the BV function
u
, only one of the following two assertions is true
lim
x
∈
Ω
(
a
^
,
x
0
)
x
→
x
0
u
(
x
)
=
lim
x
∈
Ω
(
−
a
^
,
x
0
)
x
→
x
0
u
(
x
)
lim
x
∈
Ω
(
a
^
,
x
0
)
x
→
x
0
u
(
x
)
≠
lim
x
∈
Ω
(
−
a
^
,
x
0
)
x
→
x
0
u
(
x
)
or
x
0
belongs to a subset of
Ω
having zero
n
−
1
dimensional Hausdorff measure. The quantities
lim
x
∈
Ω
(
a
^
,
x
0
)
x
→
x
0
u
(
x
)
=
u
a
^
(
x
0
)
lim
x
∈
Ω
(
−
a
^
,
x
0
)
x
→
x
0
u
(
x
)
=
u
−
a
^
(
x
0
)
are called approximate limits of the BV function
u
at the point
x
0
.
The functional
V
(
⋅
,
Ω
)
:
B
V
(
Ω
)
→
R
+
is lower semicontinuous: to see this, choose a Cauchy sequence of BVfunctions
{
u
n
}
n
∈
N
converging to
u
∈
L
loc
1
(
Ω
)
. Then, since all the functions of the sequence and their limit function are integrable and by the definition of lower limit
lim inf
n
→
∞
V
(
u
n
,
Ω
)
≥
lim inf
n
→
∞
∫
Ω
u
n
(
x
)
d
i
v
ϕ
d
x
≥
∫
Ω
lim
n
→
∞
u
n
(
x
)
d
i
v
ϕ
d
x
=
∫
Ω
u
(
x
)
d
i
v
ϕ
d
x
∀
ϕ
∈
C
c
1
(
Ω
,
R
n
)
,
∥
ϕ
∥
L
∞
(
Ω
)
≤
1
Now considering the supremum on the set of functions
ϕ
∈
C
c
1
(
Ω
,
R
n
)
such that
∥
ϕ
∥
L
∞
(
Ω
)
≤
1
then the following inequality holds true
lim inf
n
→
∞
V
(
u
n
,
Ω
)
≥
V
(
u
,
Ω
)
which is exactly the definition of lower semicontinuity.
By definition
B
V
(
Ω
)
is a subset of
L
1
(
Ω
)
, while linearity follows from the linearity properties of the defining integral i.e.
∫
Ω
[
u
(
x
)
+
v
(
x
)
]
d
i
v
ϕ
(
x
)
d
x
=
∫
Ω
u
(
x
)
d
i
v
ϕ
(
x
)
d
x
+
∫
Ω
v
(
x
)
d
i
v
ϕ
(
x
)
d
x
=
=
−
∫
Ω
⟨
ϕ
(
x
)
,
D
u
(
x
)
⟩
−
∫
Ω
⟨
ϕ
(
x
)
,
D
v
(
x
)
⟩
=
−
∫
Ω
⟨
ϕ
(
x
)
,
[
D
u
(
x
)
+
D
v
(
x
)
]
⟩
for all
ϕ
∈
C
c
1
(
Ω
,
R
n
)
therefore
u
+
v
∈
B
V
(
Ω
)
for all
u
,
v
∈
B
V
(
Ω
)
, and
∫
Ω
c
⋅
u
(
x
)
d
i
v
ϕ
(
x
)
d
x
=
c
∫
Ω
u
(
x
)
d
i
v
ϕ
(
x
)
d
x
=
−
c
∫
Ω
⟨
ϕ
(
x
)
,
D
u
(
x
)
⟩
for all
c
∈
R
, therefore
c
u
∈
B
V
(
Ω
)
for all
u
∈
B
V
(
Ω
)
, and all
c
∈
R
. The proved vector space properties imply that
B
V
(
Ω
)
is a vector subspace of
L
1
(
Ω
)
. Consider now the function
∥
∥
B
V
:
B
V
(
Ω
)
→
R
+
defined as
∥
u
∥
B
V
:=
∥
u
∥
L
1
+
V
(
u
,
Ω
)
where
∥
∥
L
1
is the usual
L
1
(
Ω
)
norm: it is easy to prove that this is a norm on
B
V
(
Ω
)
. To see that
B
V
(
Ω
)
is complete respect to it, i.e. it is a Banach space, consider a Cauchy sequence
{
u
n
}
n
∈
N
in
B
V
(
Ω
)
. By definition it is also a Cauchy sequence in
L
1
(
Ω
)
and therefore has a limit
u
in
L
1
(
Ω
)
: since
u
n
is bounded in
B
V
(
Ω
)
for each
n
, then
∥
u
∥
B
V
<
+
∞
by lower semicontinuity of the variation
V
(
⋅
,
Ω
)
, therefore
u
is a BV function. Finally, again by lower semicontinuity, choosing an arbitrary small positive number
ε
∥
u
j
−
u
k
∥
B
V
<
ε
∀
j
,
k
≥
N
∈
N
⇒
V
(
u
k
−
u
,
Ω
)
≤
lim inf
j
→
+
∞
V
(
u
k
−
u
j
,
Ω
)
≤
ε
To see this, it is sufficient to consider the following example belonging to the space
B
V
(
[
0
,
1
]
)
: for each 0<α<1 define
χ
α
=
χ
[
α
,
1
]
=
{
0
if
x
∉
[
α
,
1
]
1
if
x
∈
[
α
,
1
]
as the characteristic function of the leftclosed interval
[
α
,
1
]
. Then, choosing α,β∈
[
0
,
1
]
such that α≠β the following relation holds true:
∥
χ
α
−
χ
β
∥
B
V
=
2
+

α
−
β

Now, in order to prove that every dense subset of
B
V
(
]
0
,
1
[
)
cannot be countable, it is sufficient to see that for every α∈
[
0
,
1
]
it is possible to construct the balls
B
α
=
{
ψ
∈
B
V
(
[
0
,
1
]
)
;
∥
χ
α
−
ψ
∥
B
V
≤
1
}
Obviously those balls are pairwise disjoint, and also are an indexed family of sets whose index set is
[
0
,
1
]
. This implies that this family has the cardinality of the continuum: now, since any dense subset of
B
V
(
[
0
,
1
]
)
must have at least a point inside each member of this family, its cardinality is at least that of the continuum and therefore cannot a be countable subset. This example can be obviously extended to higher dimensions, and since it involves only local properties, it implies that the same property is true also for
B
V
l
o
c
.
Chain rules for nonsmooth functions are very important in mathematics and mathematical physics since there are several important physical models whose behavior is described by functions or functionals with a very limited degree of smoothness.The following version is proved in the paper (Vol'pert 1967, p. 248): all partial derivatives must be intended in a generalized sense. i.e. as generalized derivatives
Theorem. Let
f
:
R
p
→
R
be a function of class
C
1
(i.e. a continuous and differentiable function having continuous derivatives) and let
u
(
x
)
=
(
u
1
(
x
)
,
…
,
u
p
(
x
)
)
be a function in
B
V
(
Ω
)
with
Ω
being an open subset of
R
n
. Then
f
∘
u
(
x
)
=
f
(
u
(
x
)
)
∈
B
V
(
Ω
)
and
∂
f
(
u
(
x
)
)
∂
x
i
=
∑
k
=
1
p
∂
f
¯
(
u
(
x
)
)
∂
u
k
∂
u
k
(
x
)
∂
x
i
∀
i
=
1
,
…
,
n
where
f
¯
(
u
(
x
)
)
is the mean value of the function at the point
x
∈
Ω
, defined as
f
¯
(
u
(
x
)
)
=
∫
0
1
f
(
u
a
^
(
x
)
t
+
u
−
a
^
(
x
)
(
1
−
t
)
)
d
t
A more general chain rule formula for Lipschitz continuous functions
f
:
R
p
→
R
s
has been found by Luigi Ambrosio and Gianni Dal Maso and is published in the paper (Ambrosio & Dal Maso 1990). However, even this formula has very important direct consequences: choosing
f
(
u
)
=
v
(
x
)
u
(
x
)
, where
v
(
x
)
is also a
B
V
function, the preceding formula gives the Leibniz rule for
B
V
functions
∂
v
(
x
)
u
(
x
)
∂
x
i
=
u
¯
(
x
)
∂
v
(
x
)
∂
x
i
+
v
¯
(
x
)
∂
u
(
x
)
∂
x
i
This implies that the product of two functions of bounded variation is again a function of bounded variation, therefore
B
V
(
Ω
)
is an algebra.
This property follows directly from the fact that
B
V
(
Ω
)
is a Banach space and also an associative algebra: this implies that if
{
v
n
}
and
{
u
n
}
are Cauchy sequences of
B
V
functions converging respectively to functions
v
and
u
in
B
V
(
Ω
)
, then
therefore the ordinary product of functions is continuous in
B
V
(
Ω
)
respect to each argument, making this function space a Banach algebra.
It is possible to generalize the above notion of total variation so that different variations are weighted differently. More precisely, let
φ
:
[
0
,
+
∞
)
⟶
[
0
,
+
∞
)
be any increasing function such that
φ
(
0
)
=
φ
(
0
+
)
=
lim
x
→
0
+
φ
(
x
)
=
0
(the weight function) and let
f
:
[
0
,
T
]
⟶
X
be a function from the interval
[
0
,
T
]
⊂ℝ taking values in a normed vector space
X
. Then the
φ
variation of
f
over
[
0
,
T
]
is defined as
φ
Var
[
0
,
T
]
(
f
)
:=
sup
∑
j
=
0
k
φ
(

f
(
t
j
+
1
)
−
f
(
t
j
)

X
)
,
where, as usual, the supremum is taken over all finite partitions of the interval
[
0
,
T
]
, i.e. all the finite sets of real numbers
t
i
such that
0
=
t
0
<
t
1
<
…
<
t
k
=
T
.
The original notion of variation considered above is the special case of
φ
variation for which the weight function is the identity function: therefore an integrable function
f
is said to be a weighted BV function (of weight
φ
) if and only if its
φ
variation is finite.
f
∈
B
V
φ
(
[
0
,
T
]
;
X
)
⟺
φ
Var
[
0
,
T
]
(
f
)
<
+
∞
The space
B
V
φ
(
[
0
,
T
]
;
X
)
is a topological vector space with respect to the norm
∥
f
∥
B
V
φ
:=
∥
f
∥
∞
+
φ
Var
[
0
,
T
]
(
f
)
,
where
∥
f
∥
∞
denotes the usual supremum norm of
f
. Weighted BV functions were introduced and studied in full generality by Władysław Orlicz and Julian Musielak in the paper Musielak & Orlicz 1959: Laurence Chisholm Young studied earlier the case
φ
(
x
)
=
x
p
where
p
is a positive integer.
SBV functions i.e. Special functions of Bounded Variation were introduced by Luigi Ambrosio and Ennio de Giorgi in the paper (Ambrosio & De Giorgi 1988), dealing with free discontinuity variational problems: given an open subset
Ω
of ℝ^{n}, the space
S
B
V
(
Ω
)
is a proper linear subspace of
B
V
(
Ω
)
, since the weak gradient of each function belonging to it consists precisely of the sum of an
n
dimensional support and an
n
−
1
dimensional support measure and no intermediatedimensional terms, as seen in the following definition.
Definition. Given a locally integrable function
u
, then
u
∈
S
B
V
(
Ω
)
if and only if
1. There exist two Borel functions
f
and
g
of domain
Ω
and codomain ℝ^{n} such that
∫
Ω

f

d
H
n
+
∫
Ω

g

d
H
n
−
1
<
+
∞
.
2. For all of continuously differentiable vector functions
ϕ
of compact support contained in
Ω
, i.e. for all
ϕ
∈
C
c
1
(
Ω
,
R
n
)
the following formula is true:
∫
Ω
u
div
ϕ
d
H
n
=
∫
Ω
⟨
ϕ
,
f
⟩
d
H
n
+
∫
Ω
⟨
ϕ
,
g
⟩
d
H
n
−
1
.
where
H
α
is the
α
dimensional Hausdorff measure.
Details on the properties of SBV functions can be found in works cited in the bibliography section: particularly the paper (De Giorgi 1992) contains a useful bibliography.
As particular examples of Banach spaces, Dunford & Schwartz (1958, Chapter IV) consider spaces of sequences of bounded variation, in addition to the spaces of functions of bounded variation. The total variation of a sequence x=(x_{i}) of real or complex numbers is defined by
T
V
(
x
)
=
∑
i
=
1
∞

x
i
+
1
−
x
i

.
The space of all sequences of finite total variation is denoted by bv. The norm on bv is given by
∥
x
∥
b
v
=

x
1

+
T
V
(
x
)
=

x
1

+
∑
i
=
1
∞

x
i
+
1
−
x
i

.
With this norm, the space bv is a Banach space.
The total variation itself defines a norm on a certain subspace of bv, denoted by bv_{0}, consisting of sequences x = (x_{i}) for which
lim
n
→
∞
x
n
=
0.
The norm on bv_{0} is denoted
∥
x
∥
b
v
0
=
T
V
(
x
)
=
∑
i
=
1
∞

x
i
+
1
−
x
i

.
With respect to this norm bv_{0} becomes a Banach space as well.
A signed (or complex) measure
μ
on a measurable space
(
X
,
Σ
)
is said to be of bounded variation if its total variation
∥
μ
∥
=

μ

(
X
)
is bounded: see Halmos (1950, p. 123), Kolmogorov & Fomin (1969, p. 346) or the entry "Total variation" for further details.
The function
f
(
x
)
=
{
0
,
if
x
=
0
sin
(
1
/
x
)
,
if
x
≠
0
is not of bounded variation on the interval
[
0
,
2
/
π
]
While it is harder to see, the continuous function
f
(
x
)
=
{
0
,
if
x
=
0
x
sin
(
1
/
x
)
,
if
x
≠
0
is not of bounded variation on the interval
[
0
,
2
/
π
]
either.
At the same time, the function
f
(
x
)
=
{
0
,
if
x
=
0
x
2
sin
(
1
/
x
)
,
if
x
≠
0
is of bounded variation on the interval
[
0
,
2
/
π
]
. However, all three functions are of bounded variation on each interval
[
a
,
b
]
with
a
>
0
.
The Sobolev space
W
1
,
1
(
Ω
)
is a proper subset of
B
V
(
Ω
)
. In fact, for each
u
in
W
1
,
1
(
Ω
)
it is possible to choose a measure
μ
:=
∇
u
L
(where
L
is the Lebesgue measure on
Ω
) such that the equality
∫
u
d
i
v
ϕ
=
−
∫
ϕ
d
μ
=
−
∫
ϕ
∇
u
∀
ϕ
∈
C
c
1
holds, since it is nothing more than the definition of weak derivative, and hence holds true. One can easily find an example of a BV function which is not
W
1
,
1
: in dimension one, any step function with a nontrivial jump will do.
Functions of bounded variation have been studied in connection with the set of discontinuities of functions and differentiability of real functions, and the following results are wellknown. If
f
is a real function of bounded variation on an interval
[
a
,
b
]
then
f
is continuous except at most on a countable set;
f
has onesided limits everywhere (limits from the left everywhere in
(
a
,
b
]
, and from the right everywhere in
[
a
,
b
)
;
the derivative
f
′
(
x
)
exists almost everywhere (i.e. except for a set of measure zero).
For real functions of several real variables
the characteristic function of a Caccioppoli set is a BV function: BV functions lie at the basis of the modern theory of perimeters.
Minimal surfaces are graphs of BV functions: in this context, see reference (Giusti 1984).
The ability of BV functions to deal with discontinuities has made their use widespread in the applied sciences: solutions of problems in mechanics, physics, chemical kinetics are very often representable by functions of bounded variation. The book (Hudjaev & Vol'pert 1985) details a very ample set of mathematical physics applications of BV functions. Also there is some modern application which deserves a brief description.
The MumfordShah Functional: the segmentation problem for a twodimensional image, i.e. the problem of faithful reproduction of contours and grey scales is equivalent to the minimization of such functional.